Embed Size (px)
Citation preview
No. 2003–71
Sustainability and substitution of exhaustible natural resources How resource prices affect long-term R&D-investments
By Lucas Bretschger and Sjak Smulders
July 2003
ISSN 0924-7815
Sustainability and substitution of exhaustible natural resourcesHow resource prices affect long-term R&D-investments
Lucas Bretschger*andSjak Smulders**
July 2003
AbstractTraditional resource economics has been criticised for assuming too high elasticitiesof substitution, not observing material balance principles and relying too much onplanner solutions to obtain long-term growth. By analysing a multi-sector R&D-based endogenous growth model with exhaustible natural resources, labour,knowledge, and physical capital as inputs, the present paper addresses this critique.We study transitional dynamics and the long-term growth path and identifyconditions under which firms keep spending on research and development. Wedemonstrate that long-run growth can be sustained under free market conditionseven when elasticities of substitution between capital and resources are low and thesupply of physical capital is limited, which seems to be crucial for today’ssustainability debate.
Keywords : Growth, non-renewable resources, substitution, investmentincentives, endogenous technological change, sustainability
JEL-Classification : Q20, Q30, O41, O33
* Corresponding author: WIF – Institute of Economic Research, ETH-Zentrum, WETD5, CH-8092 Zürich, [email protected].** Department of Economics, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, TheNetherlands, [email protected].
We thank Christian Groth, Karl-Josef Koch, Peter Kort, Thomas Steger, Cees Withagen,and various seminar and session participants for valuable comments. Smulders'research is supported by the Royal Netherlands Academy of Arts and Sciences.
1
1. IntroductionSteady accumulation of man-made capital is the basic source of economic growth.Capital investments dropped substantially during the high-price period on marketsfor non-renewable resources in the 1970s. Since then, known stocks of resources haveincreased due to discoveries, prices have moderated, and investments haverecovered. But given the finiteness of resources such as fossil fuels and preciousminerals in the long run, will resource scarcity limit the future level of capitalaccumulation? This question is crucial in the context of sustainability, because, tomaintain or increase consumption, capital has to substitute for resources inproduction. As soon as investments become critically low, future generations’welfare levels fall behind those of today’s generations, which means that long-termdevelopment no longer satisfies the sustainability criterion.
Limited substitution of man-made capital for non-renewable natural resourcesmay be the main obstacle to sustainable development. Most ecological economistsargue that traditional economic theories are overly optimistic in this respect. Threemajor issues are under debate. First, to obtain unbounded growth in the standardneoclassical model, it has to be assumed that either the elasticity of substitutionbetween natural resources and man-made capital is at least unity, or that exogenousresource-augmenting technological progress occurs at a constant rate, see the seminalpapers of Dasgupta and Heal (1974) and Stiglitz (1974). Second, while manyeconomic models rely on unlimited accumulation of man-made capital, ecologicaleconomists emphasise that material balances limit the use of physical capital in thelong run, see Cleveland and Ruth (1997). Third, while sustained growth may betechnically feasible, it is not necessarily reached under free market conditions. Lowinvestment incentives and externalities may result in (too) little investment efforts incapital that substitutes for resources. Moreover, myopic behaviour may prevent theimplementation by today’s generations of policy measures that are needed to obtainsustainability for future generations.
The present paper reconsiders investment incentives and the limits to growth inthe presence of non-renewable resource scarcity in a multi-sector endogenous model.We rule out exogenous technological change that offsets resource depletion as“manna from heaven”, we bound the total supply of physical capital to take intoaccount material balances, and we concentrate on market equilibria in which rates ofreturn drive investment in physical capital and research and development. We showthat growth can be sustained even if elasticities of substitution are small. The multi-sector structure of the model allows us to identify how substitution between sectorsworks as an additional mechanism for the substitution of natural resources. Inparticular, it will be shown that the effects on growth depend on which sector of theeconomy has poor or abundant substitution possibilities.
We focus on investment in knowledge capital through research anddevelopment (R&D) as the main engine of growth. R&D is often aimed at improvingproduction techniques and thus at increasing capital productivity. For a large part ofthe economy, knowledge is only effective if it is embodied in certain types of physicalcapital. The decisive question, which is not answered yet in literature, reads: is itrealistic to predict that knowledge accumulation is so powerful as to outweigh thephysical limits of physical capital services and the limited substitution possibilitiesfor natural resources? Only if the answer is positive the economy can provide a
2
constant or steadily increasing level of average individual utility in the long run,which is required for development to become sustainable. To eliminate hypotheticaltechnical solutions for substitution, we focus on market outcomes to find outwhether market incentives are strong enough to produce a sustainable level ofinvestments in knowledge capital.
The present paper builds on various contributions in literature. The neoclassicalliterature on resource economics started with the seminal symposium issue of theReview of Economic Studies, 1974. This tradition stresses that the marginal returnson capital (and thus investment incentives) decrease when substitution possibilitiesbetween physical capital and exhaustible natural resources are low. In particular,Dasgupta and Heal (1974, 1979) show that, without technical progress and with anelasticity of substitution between natural resources and man-made capital beinglower than unity, sustainable production is impossible. Indeed, if the elasticity ofsubstitution is zero, which is implicitly assumed in the popular contribution ofMeadows et al. (1972), the economic collapse is inevitable. Furthermore, Solow (1974)derives that sustainable production may be feasible without technical progress,whenever the elasticity equals unity. Provided that the production elasticity ofcapital exceeds the elasticity of natural resources, an appropriate constant share ofincome spent for savings leads to a constant flow of income, see Hartwick (1977).However, as soon as savings depend on interest rates, this result will not be obtainedunder free market conditions. Hence, an equilibrium with non-declining income isneither optimal nor does arise in a market equilibrium.
Taking into account technical change leads to less pessimistic results. In Stiglitz(1974), exogenous technical progress leads to sustained growth, which is feasible andoptimal, provided that the elasticity is unity and the discount rate is not too high.Introducing exogenous resource-augmenting technical progress, sustained growthbecomes feasible even when the elasticity is lower than unity; see Dasgupta and Heal(1979, p. 207). The optimistic view on technology in neoclassical models is criticisedby Cleveland and Ruth (1997), who argue that in reality substitutability is low, thatthe continuity of technical progress is uncertain, and that the accumulation ofphysical capital is ultimately limited by biophysical constraints. Under theseassumptions, sustainability indeed becomes a more demanding goal.
Long-run growth does not rely on exogenous technological change in the so-called new (or endogenous) growth theory developed in the 1990s. The endogenousaccumulation of knowledge and human capital supplements the accumulation ofphysical capital as an engine of growth. The broadened view of man-made capital,the hypothesis of positive knowledge spillovers, and the assumed constant returns toaggregate capital inputs provide new perspectives on substitution of man-madecapital for natural resources and sustained investment incentives.
For the case a unitary elasticity of substitution between capital and resourceinputs, endogenous knowledge accumulation yields sustained optimal growth in thepresence of non-renewable resources (Schou 1999, Scholz and Ziemes 1999, Aghionand Howitt 1998, Grimaud and Rougé 2003, Groth and Schou 2002). In the two-sectorendogenous growth model with resource-augmenting knowledge production ofBovenberg and Smulders (1995), unbounded growth is feasible with elasticities ofsubstitution between capital and resources that are smaller than unity. However, themodel only applies to renewable resources. This is also true for Bretschger (1998),
3
who shows that lower resource use is compatible with sustained endogenous growthunder free market conditions even when technical change is unbiased and theelasticity of substitution in production is smaller than unity.
The model of the present paper has the following special features.First, we distinguish between two types of physical capital and knowledge
capital. Physical capital stocks are a direct (but limited) substitute for naturalresources whereas knowledge capital is directed at improving the efficiency ofcapital in one of the two consumer sectors. Accordingly, we call this consumer sectorthe “knowledge-using” sector. It is assumed that new ideas need to be embodied in acertain physical body before they can be used as a substitute for a natural resource.The heterogeneity of physical capital opens the possibilities of productivity gains inthe knowledge-using sector by increasing division of labour. Applying the idea ofexpansion-in-varieties, this part of the model builds on Romer (1990) and Grossmanand Helpman (1991). The increasing division of labour is assumed to have a similareffect on the efficiency of natural resource use, which is a scale effect that has to bedistinguished from the basic substitution effect. In the other consumer sector, capitalproductivity depends on the input of skilled labour. As skilled labour is also animportant input into R&D, which produces knowledge through positive spillovers,this sector is called “knowledge-competing” sector. Deviating from unitaryelasticities, it will become evident that in the knowledge-using sector, a highelasticity of substitution fosters economic growth. In the knowledge-competingsector, however, a low elasticity is more favourable for long-term development.
Second, we introduce a non-renewable resource, which represents oil, preciousmetals, minerals etc. The resource is used in combination with physical capital toproduce the two final outputs in the knowledge-using and the knowledge-competingsector. Natural resources are thus important for both sectors. Accordingly, a double-tracked input substitution process is modelled below.
Third, we emphasise the distinction between the market value and theproduction cost of additional designs. This alludes to Tobin’s q-theory of investment.Profit expectations in research are a variable that explicitly depends on the price ofphysical capital and, indirectly, on the price of the natural resource. On the otherhand, costs of inventions are determined in a separate R&D sector, where skilledlabour and knowledge are used as inputs. As skilled labour is also used incombination with the natural resource to produce final output, production cost in theresearch lab depend on resource prices.
Fourth, we do not exclusively focus on balanced growth paths but also look atadjustment paths leading to long-run equilibria. During adjustment, goods andfactor prices as well as relative sectoral outputs are allowed to vary. It is shown howthe size of the sectors and the growth rate converge to long-run values according tothe assumed parameters for substitution.
The remainder of the paper is organized as follows. In section 2, the theoreticalfive-sector model of the economy with two consumer goods is presented in detail.Section 3 shows how the model can be solved. Section 4 provides results fortransitional dynamics and long-run growth for different types of parameter andsubstitution conditions. Section 5 concludes.
4
2. The model
2.1 OverviewWe introduce three primary input factors: a non-renewable natural resource R aswell as skilled labour S and unskilled labour L, see figure 1. Unskilled labourproduces differentiated physical capital components, which are assembled to anaggregate physical capital stock YK . Skilled labour is employed in two activities.First, the development of designs for new capital components requires skills. Second,skilled labour produces the physical capital good TK . The natural resource iscombined with the two capital stocks and produces “standard” T-goods in theknowledge-competing sector and “high-tech” Y-goods in the knowledge-usingsector. The substitution of capital for the natural resource takes place in both sectors:in the high-tech sector, which determines the reward to research investments, and inthe standard sector, which affects wages of skilled labour and therefore productioncosts of new designs. Knowledge capital is accumulated through positive spilloversin research and is an input into subsequent R&D; it is the driving force for long-rundevelopment.
The essential elements of our model set-up are, first, substitution acrosssectors between goods that differ in their knowledge intensity and, second, (poor)substitution between man-made inputs and resources within sectors. The degree ofwithin-sector substitution may differ across sectors. By combining these differentmechanisms, the model is suited to capture the basic substitution process in moderneconomies.
Fig. 1(about here)
2.2 Production sectorLet us discuss the different productive sectors of the model in turn, beginning withhigh-tech goods, followed by standard goods, the two types of capital goods, and,finally, R&D.
High-tech goods Y are produced with different physical capital services k andnatural resources R as inputs. We adopt a nested CES-function. The constantelasticity of substitution between aggregate capital and natural resources σ in the Y-sector take any positive value. Aggregate capital input is a CES-index of a continuumof differentiated components of mass n; the constant elasticity of substitutionbetween them equals 1/(1 – β)>0. Each producer in the high-tech sector uses all typesof components as well as natural resources, according to:
( ) ( )1 11
0(1 )
n
j YY k dj n R
σσ σσβσβ δ σθ θ− −−
= + − ⋅ ∫ , (1)
5
where 0 , 1β θ< < , and , 0δ σ > are given parameters. In a symmetrical equilibrium,the quantities of capital services k are equal for the different components, i.e.
1 2 ... nk k k k= = = = . With n different components at a certain point of time, aggregateinput of capital services is denoted by
YK n k= ⋅ . (2)
Due to gains from specialization, an expansion-in-varieties of capital componentsleads to productivity gains of Y-producers. According to (1), holding aggregatecapital input YK constant, the production of the high-tech goods increases with thenumber of capital components, n. In a similar way, it is reasonable to argue that Y-producers are able to use natural resources the more efficiently, the higher is thespecialization of capital components a given amount of resources is combined with.We will come back to this point when discussing the general results at the end ofsection 4. The gains from specialization for both types of inputs can be seen moreclearly when reformulating (1) by using (2) to get:
( )1 1
1 1
(1 )Y YY n K n R
− −− −
= ⋅ + − ⋅
σσ σ
β σσδβ σθ θ .
The specialization effect of an additional capital variety is given by (1−β)/ β whereasthe efficiency gain of resources R used with additional varieties is expressed by thefactor δ . If 0δ = , there is no efficiency gain from resource use. Rearranging theabove equation, we find:
( ) /( 1)1 /(1 ) / (1 ) / ( 1) / (1 )( )Y YY n Q n K n R−−− − − − = ⋅ = ⋅ + −
σ σσ σβ β β β σ σ νθ θ (3)
with (1 ) /= − −ν β β δ . The expression in brackets on the r.h.s. of (3) corresponds tothe familiar CES-approach of resource economics, see Dasgupta/Heal (1979, p. 199).It aggregates capital inputs YK and effective resource inputs Yn R−ν into a compositeinput Q. The total factor productivity term preceding brackets is the result of theexpansion-in-varieties approach according to Romer (1990) and Grossman/Helpman(1991). Note that the degree of specialization (measured by n) affects productivity intwo ways. First, it raises total factor productivity. Second, it introduces a bias intechnological change: if (1 )−σ ν < 0 technological change is capital-using; if (1 )−σ ν> 0, it is resource-using.
The market for Y-goods is fully competitive. Producers take prices of output,resource inputs and capital components (denoted by ,Y Rp p and kjp , respectively) as
given. They maximize total profits 0
n
Y R Y kj jp Y p R p k dj− −∫ , subject to the production
function (1). Under symmetry ( kj kYp p= ), see (2), relative demand for capital andenergy is given by:
6
(1 )
1kYY
Y R
pK nR p
−− −
= −
σ σσ νθ
θ. (4)
The production of standard goods T requires homogenous capital TK andresources TR as inputs. We use again a CES-formulation:
( )1 1 11T TT K R
ωω ω ωω ωη η− − − = ⋅ + −
(5)
with ω being the elasticity of substitution in the T-sector and 0 1η< < . Producerstake prices as given and maximize profits T KT T R Tp T p K p R− − subject to (5). Thisgives relative factor demand:
1T KT
T R
K pR p
− = −
ω ωηη
(6)
The differentiated capital services k are produced by monopolists. Theproduction of one unit requires one unit of unskilled labour L. The profit maximizingmonopolistic supplier faces a price elasticity of demand equal to –1/(1 – β). As in thestandard Dixit-Stiglitz approach, this follows from the Y-producers demand for k.Thus, the monopolist optimally sets a rental rate that is a mark-up 1/β times thelabour costs wL. All monopolistic suppliers set this same price:
/kY Lp w= β (7)
Associated profits π for each supplier of a capital components can be calculated as:
(1 ) / kY Yp K n= − ⋅ ⋅π β . (8)
Profits are used to cover the expenses for fixed costs in the production of k-goods,which consist of payments for the blueprint of the capital component. Each designcontains the know-how for the production of one capital component k. Thus, each k-firm has to acquire one design as an up-front investment before it can startproduction.
New blueprints n! are produced in the R&D sector. Per blueprint, a/n units ofskilled labour are required. Here it is assumed that an increase in variety alsoincreases the stock of public knowledge on which R&D builds so that research costdecline with n. Thus, the cost of a blueprint, cn, equals:
( / )n Sc a n w= ⋅ (9)
The production of one unit of homogenous capital TK used in the T-sector requiresskilled labour TS and materials inputs M according to a Cobb-Douglas production
7
function T TK S Mυ µ= . We assume that materials are fully recyclable, such that eachmoment in time a fixed stock, of which we normalize the size to M = 1, is available.For simplicity, we disregard recycling costs, set 1=υ , and assume perfectcompetition. The price of capital then equals the wage of skilled labour Sw :
KT Sp w= . (10)
2.3 Capital marketsThere are three assets i.e. investment possibilities in the economy: riskless bonds,patents for the production of capital varieties k, and natural resources R. Let npdenote the market price of a patent in time t and consider the brief time intervalbetween t and t+dt. In this time, the return for an investment of size np in a bond is
nr p dt⋅ ⋅ . A firm holding a patent for the production of an intermediate capital inputearns an infinite stream of profits. Per-period profit π is given in (8). So in the sametime interval, the total return on a patent is ndt p dt⋅ + ⋅!π which yields the followingno-arbitrage condition:
n np r p+ = ⋅!π (11)
Perfect capital markets affect the timing of R&D. Over a small period dt with positiveinvestment in R&D, inventors should be indifferent between incurring the R&D cost
nc at the beginning or the end of the period. That is, the return to postponinginvestment, which amounts to interest payments on postponed costs nrc dt , shouldequal the cost of postponing, which amount to forgone dividends and change( )nc dtπ+ ⋅! . However, if over a small period the net benefits of postponinginvestment are positive, no investment will take place in this period. Hence, we maywrite the following no-arbitrage condition:
n nc r c+ ≤ ⋅!π with equality (inequality) if 0n >! ( 0n =! ). (12)
The final no-arbitrage condition concerns the comparison of returns betweenbonds and stocks of natural resources. Resource owners extract resources withoutcosts and supply them at spot prices, which are denoted by Rp per unit of R. Thereturns for investments of size Rp in bonds during the brief time interval between tand t+dt are Rr p⋅ . In analogy to above, the return per unit of the stock of naturalresource R can be expressed as R Rdt p dtπ ⋅ + ⋅! . However, it is a basic characteristic ofnatural resources to have no direct return like capital goods, i.e. it is that 0Rπ = . Soin equilibrium we are left with:
R Rp r p= ⋅! (13)
The Hotelling-rule in (13) implies that resource owners are exactly indifferentbetween selling resources (and investing the profit with interest rate r ) andpreserving the stock of resources. The compensation for keeping the stock is the pricerise of the resource.
8
2.4 Factor marketsThe total stock of resource R at time t is denoted by WR . It is depleted according to:
R Y TW R R= − −! , (0)RW given, ( ) 0RW t ≥ , (14)
( )0
( ) ( ) (0)∞
+ =∫ Y T RR t R t dt W , (15)
which says that any flow of resource use depletes the total resource stockproportionally, that the resource stock is predetermined, and that the stock can neverbecome negative. Profit maximization by resource owners implies that the price ofnatural resources increases at a rate equal to the interest rate, see (13). Resourceowners do not have an incentive to conserve a part of the stock so that totalextraction must be equal to the total resource stock in equilibrium, see (15). Totalextraction can be ensured by setting the optimum price at the beginning ofoptimisation. Initial price level and price increase of natural resources do not deviatefrom the optimum path, provided that agents form rational expectations. This will beassumed in the following.
The market for skilled labour is in equilibrium if the fixed supply S equalsdemand for production of capital and for research labour:
( / )TS K a n n= + ⋅ ! (16)
The market for unskilled labour clears, which requires that demand for unskilledlabour from the differentiated capital goods sector equals the fixed supply L:
YL K= . (17)
2.5 Consumer sectorThe representative household maximizes a lifetime utility function subject to theusual intertemporal budget constraint; the function is additively separable in timeand contains logarithmic intratemporal utility of the Cobb-Douglas type:
τττρ dCetUt
t )(log)( )(∫∞ −−= → max (18)
with 1C Y Tφ φ−= ⋅ (19)
The Cobb-Douglas specification in (19) implies constant expenditure shares for T-and Y-goods. Accordingly, relative demand for final goods is given by:
1Y
T
p Yp T
φφ
⋅ =⋅ −
. (20)
9
where (, )T Yp p is the price of T-goods (Y-goods). Intertemporal optimisation givesthe well-known Keynes-Ramsey rule stating that the growth rate of consumerexpenditures is equal to the difference between the nominal interest rate r and thediscount rate ρ , that is:
ˆˆTp T r+ = −ρ . (21)
where hats denore growth rates.Finally, intertemporal utility maximization requires that the value of
household wealth, npn, discounted by the interest rate, r, approaches zero if time goesto infinity. This transversality condition can be written as:
ˆ ˆlim ( ) ( ) ( ) 0t nn t p t r t→∞ + − < . (22)
3. Solving the model
To facilitate the analysis, we reduce the model to three differential equations, whichcharacterize the dynamics of the system. Our three key variables will be the capitalshares in the knowledge-using and knowledge competing sectors, to be denoted by θand η respectively, and the growth rate of product variety, also to be interpreted asthe rate of innovation or the growth rate of the (public) knowledge stock, to bedenoted by g. Hence, we define:
kY Y
Y
p Kp Y
=θ (23)
KT T
T
p Kp T
=η (24)
ˆg n= . (25)
Note that the initial knowledge stock, n(0), is given. Because total revenue equalstotal cost for energy and capital inputs, the energy shares in the two sectors are givenby 1 /R T Tp R p T− =η and 1 /R Y Yp R p Y− =θ , respectively. Inserting the definitions (23) and(24) into the firms demand functions (4) and (6), differentiating (4) and (7) with respect totime and using (7), (10) and (13) to eliminate capital and resource prices, we find:
ˆ ˆ(1 )(1 )( )Lr w g= − − − − +θ θ σ ν , (26)
ˆ ˆ(1 )(1 )( )Sr w= − − − −η η ω . (27)
10
The equations show how interest cost, wage changes, and technical change (only inthe Y-sector) drive factor substitution.
Substituting (23) and (24) into (19) to eliminate Yp Y and Tp T and inserting (7)and (17), we find:
1L T
S
w Kw L
θ βφη φ
= ⋅−
. (28)
This equation reflects that in final-goods market equilibrium the relative wage ofunskilled labour increases with the share of capital in the unskilled-labour-intensiveY-sector (θ), increases with the share of the Y-good in total demand for final goods(φ), and falls with the relative abundance of unskilled labour.
Substituting ˆ ˆˆ ˆ ˆT S Tp T w K+ = + −η , which follows from (24) and (10), into (21),we find:
ˆˆ ˆS Tr w Kρ η− = + − . (29)
Substituting (25) into (16), we express KT in terms of the innovation rate:
TK S ag= − . (30)
From (28), (29) and (26) and from (29), (30) and (27), we derive the followingequations of motion for the cost shares, respectively:
(1 )(1 ) ( )1 (1 )(1 )
g − −= − + − − − ! σ θθ θ ρ ν
σ θ(31)
(1 )(1 )1 (1 )(1 ) /
gS a g
− −= − − − − − −
!! ω ηη η ρω η
(32)
Dividing the capital market equilibrium (12) by the R&D cost cn, inserting (7),(8), (9), (17) and (25), we find:
1ˆ LS
S
wLr w ga w
−− ≥ −ββ
. with equality for 0g > (33)
This right-hand side of (33) represents the rate of return to R&D. It increases with themark-up rate 1/β and with the size of the market as captured by L, which is thelabour supply that produces the stock of physical capital in which innovations areembodied. The rate of return falls with the cost of R&D, which is proportional to awS.
Substituting (30), (28), (27) and (32) into (33), we find:
11
[ ]1 (1 )(1 )S Sg g g ga a
= − − − − −Φ − + ! θω η ρ
η, for g>0, (34)
1 (1 )(1 )Sa
ρ θ> Φ− −ω − η η
, for 0g g= =! , (35)
with (1 )1β φφ
−Φ =−
.
Equations (34), (32) and (31) now form a dynamic system in three variables,which are the rate of innovation g, the share of capital in Y-goods production θ , andthe share of capital in T-goods production η . None of these three variables ispredetermined, so we need to formulate the conditions that restrict initial values andend values of them. Crucial is that cumulative extraction cannot exceed the availableresource stock, see (13) and (14). We therefore need to focus on extraction andexpress it in terms of our key variables g, η and θ. Eliminating prices between (4) and(23) and between (6) and (24), and combining (23), (24) and (20), we find,respectively:
/(1 )
1 1YR Lnσ −σ
ν θ θ= −θ − θ (36)
/(1 )
( )1 1TR S ag
ω −ω η η= − − η − η
(37)
11 1
Y
T
RR
− θ Φ=− η −β
(38)
After choosing initial values for θ, η , and g, equations (36), (37) and (38), togetherwith the given initial value n(0) and equations (25), (32), (31), (34), and (35), allow usto calculate the extraction path. The initial values for θ, η , and g must be chosen suchthat the no-depletion condition (15) holds and the transversality condition (22) holds.In a steady state without innovation, ˆlim ( ) 0t n t→∞ = , discounted stock prices changeat rate ˆ / 0n np r p− = −π < , see (11). The transversality condition (22) then alwaysholds. In a steady state with innovation ( lim ( ) 0t g t→∞ > ), stock prices equal the cost ofinnovation, /n Sp aw n= , see (9), (11) and (12). Then, the transversality condition (22)boils down to:
ˆlim ( ) ( ) 0t Sr t w t→∞ − > for lim ( ) 0t g t→∞ > (39)
For future use, we express sectoral depletion rates in both final goods sectorsin terms of the three key variables. Differentiating (36) and (37) with respect to timeand substituting (31) and (32), we find, respectively:
12
[ ]1ˆ (1 )1 (1 )(1 )YR g− = + −− − −
σρ ν σ θσ θ
(40)
1ˆ (1 )1 (1 )(1 ) /T
gRS a g
− = + − − − − −
!ωρ ω η
ω η(41)
4. Solutions for different substitution conditions
To see the different mechanisms in the model most clearly, it is useful to firstconsider three specific cases for parameter values which are obtained by settingeither elasticity of substitution or both elasticities equal to one. After that, the generalcase is evaluated in section 4.4.
4.1 Cobb-Douglas CaseWe first study unitary elasticities both in the knowledge-using and the knowledge-competing sector, i.e. 1σ ω= = . From (32) and (31), we see that θ and η are constant;(36) and (37), or equivalently (4) and (7), reveal that they equal the parameters θ andη , respectively. Then the dynamics is represented by a single differential equationfor g, given by (34), which can now be simplified as:
( ) 1S Sg g ga a
θ θ ρη η
= − +Φ − Φ − ! if S
aθΦ ≥ ρη
0g = if Sa
θΦ <ρη
The corresponding phase diagram is drawn in figure 2 for the case ( / )( / )S aΦ θ η >ρ .The path converging to a negative growth rate must be ruled out. The same holds forthe path converging to /g S a= , since it violates the transversality condition (39)(first substituting ˆ 0η = , (30) and (34) into (29), and then setting /g S a= , we find
ˆ / 0Sr w S a− = − < ). Hence, the equilibrium growth rate jumps to the value for which0g =! and remains there. Thus, the equilibrium growth equals:
( / )( / )max {0, }1 ( / )
S ag Φθ η −ρ=+ Φθ η
(42)
The rate of innovation is stimulated by a higher supply of skilled labour S, a lowerunit input coefficient research a, and a lower discount rate ρ . This corresponds to thefindings in other R&D-models. Our multi-sector model reveals ho innovationincentives depend on the expenditures shares and factor shares. In particular, therate of innovation increases with /Φθ η , which captures three effects. First, sinceinnovation takes place in the Y-sector only, a higher expenditure share on Y-goods(Φ ) boosts innovation. Second, since innovation is embodied in physical capital
13
goods in the Y-sector, a greater role for capital, as measured by a larger capital sharein the Y-sector θ , increases the market for innovations and boosts research.Alternatively, a high value for θ implies a low share of non-renewable resources inY-production: the sector is less dependent on non-man-made inputs and thisstimulates innovation. Finally, and most important, innovation is high when theshare of non-renewable resources in the T-sector is high (low η ). If the T-sector reliesheavily on resources rather than skilled labour input, less skilled labour is allocatedin this sector, and more becomes available for the research sector. Hence, greaternatural-resource dependence in the knowledge-competing sector reduces output inthis sector, but raises innovation.
Fig. 2(about here)
To study how resource dependence is related to growth of consumption rather thaninnovation, we need to calculate output growth in both final goods sectors. Besidesinnovation, only depletion of resource inputs drives growth, since labour andmaterials inputs are constant. The rate of depletion equals the discount rate( ˆ ˆ
Y TR R− = − =ρ ), see (40)-(41). Differentiating consumption and production functions(19), (3) and (5) with respect to time and substituting (40)-(41), we obtain theconsumption growth rate according to:
( ) ( )ˆ 1 / 1 (1 ) (1 )(1 )C g = − + − − − + − − θ β β θ δ φ φ θ φ η ρ . (43)
Consumption grows at a positive rate only if the right-hand side of (43) is positive:innovation (at rate g, see first term at right-hand side) has to be sufficient to offset thedecline in resource inputs (at rate ρ, see second term) and to overcome the constancyof physical capital inputs. For a given rate of innovation (g), consumption growth isthe bigger, the higher are the gains from specialisation (low β ), the larger areproductivity spillovers (δ), the lower is the discount rate, and the higher are thefactor shares θ and η . A lower discount rate (ρ) reduces resource depletion andimplies a smaller drag on growth from the scarcity of non-renewable resources. Fora given rate of innovation g, a higher capital share in both sectors (θ and η ) implysmaller dependence of production on non-man-made scarce resource inputs, whichis good for growth.
Overall, resource dependence in the knowledge-competing sector (asmeasured by 1-η ) has an ambiguous impact on growth. First, higher resourcedependence makes T-goods more expensive and shifts skilled labour to innovationactivities, which increases growth through innovation (see (42)). However, higherresource dependence implies that the decline in necessary resource inputs inproduction weighs more heavily, which reduces consumption growth (see (43)).Substituting (42) into (43) and differentiating growth with respect to η , we find:
2ˆ 1 1 10 1 0(1 )
C Sa
∂ − −< ⇔ + − + < ∂ −
φ φη η τη θ β φ φ ρ
,
14
where τ stands for the expression in the first brackets in (43) and represents theeffect of innovation on output growth in the Y-sector. The inequality reveals that for
(1 ) /−η φ φ sufficiently small, higher resource dependence in the knowledge-competing sector goes together with higher growth rates.
We finally need to solve for the initial levels. Initial resource use is calculatedby using the fact that the rate of extraction R decreases with the discount rate (see 40and 41) and cumulative extraction corresponds to total resource stock. This gives
(0) (0) (0)T Y RR R W+ = ρ ⋅ . For a given initial knowledge stock n(0), we can calculate theinitial consumption and income levels. Note that a change in the initial knowledgestock, n(0), has no effect on initial factor shares and the innovation rate. The reason isthat with a Cobb-Douglas production function, technological change is neutral withrespect to production factors and affects levels of output without changing relativeprices.
4.2 Poor substitution in the knowledge-using sectorThe assumption of a unitary elasticity of substitution in the knowledge-competingsector, i.e. setting 1=ω , reduces the model to a two-dimensional system in g and θ ,given by the differential equations:
( ) 1S Sg g ga a
= − +Φ − Φ −
! θ θ ρ
η η,
(1 )(1 )( )1 (1 )(1 )
g − −= − + ⋅ ⋅− − −
! σ θθ ρ ν θσ θ
.
The corresponding phase diagram is depicted in figure 3. We have drawn thediagram assuming that ν is positive, so that the 0=!θ locus appears in the negativequadrant. If µ is positive, but sufficiently small to prevent the 0=!θ locus to intersectthe 0g =! locus, the dynamics will be qualitatively the same. Any path converging tog=S/a violates the transversality condition and must be ruled out. Any path that hitsthe g=0 line at /a Sθ > η ρ Φ must also be ruled out since it violates (35). It can be seenthat there is a unique trajectory that neither does violate the transversality condition(39) nor condition (35). This saddle path lies below the 0g =! locus. The economyjumps on the saddle path and asymptotically approaches the equilibrium with
0g= =θ .Which point on the saddle path is the equilibrium for a given initial resource
and knowledge stock, is determined by (15), (36) and (38). The saddle path defines θas a function of g, say ( ) ( ( ))θ =t f g t . Substituting this and η = η into (37) and (38), wefind aggregate resource use, RT(t)+RY(t), as a function of g(t). Since we know theequation of motion for g, this solves for the entire extraction path. In equilibrium, g(0)must be such that cumulative extraction over the entire horizon exactly equals theinitial resource stock WR(0). It now follows that a higher initial resource stock impliesan initial point further to the right on the saddle path. So a (sufficiently large)
15
resource boom boosts short-run growth. A higher initial knowledge stock increasesdepletion for given θ and g, see (36). Hence to prevent running out of resources, ahigher knowledge stock implies higher resource prices, and a lower initial growthrate in equilibrium.
During the adjustment, the rate of innovation g gradually falls to zero andthen remains zero; the share of capital θ steadily falls. With rising resource prices andpoor substitution in the Y-sector, compensation for R&D-investments is steadilyfalling. Skilled labour moves from the R&D to the T-sector. From the phase diagramit is clear that innovation stops when θ reaches the level that is implied by theintersection between the 0g =! locus and the g = 0 axis, /a Sθ η ρ= Φ . This means thatin finite time, resource prices reach such a high level that R&D becomes unprofitable.From then on, all skilled labour is in the knowledge-competing sector, knowledgegrowth is zero, and the growth rate of consumption is negative because of resourcedepletion.
Fig. 3(about here)
The main conclusion from this case is that poor substitution in the knowledge-usingsector is unambiguously unfavourable for innovation and growth.
4.3 Poor substitution in the knowledge-competing sectorAssuming a unitary elasticity for the knowledge-using sector, i.e. setting 1σ = so that=θ θ , see (31) and (36), the model reduces to a two-dimensional system in g and η ,
that reads, according to (34) and (32):
[ ]( ) 1 1 (1 )(1 )1 (1 )(1 )
S Sg g ga a
θ θ ρ ω ηη η ω η
= − +Φ − Φ − − − − − − − ! ,
(1 )(1 ) 1 Sga
θ θη η ω ηη η
= − − +Φ −Φ ! .
The corresponding phase diagram is depicted in figure 4. Any path converging toη=1 must be ruled out since it violates the transversality condition (it implies ˆ 0η > ,so that ˆ 0Sr w− < , see (27), which violates (39)). Any path converging to g=0 and η=0must also be ruled out since it violates (35). Hence, the economy jumps on the saddlepath, which lies between the 0η =! and 0g =! loci, and asymptotically approaches theequilibrium with 0η = and g = S/a.
Which point on the saddle path is the equilibrium for a given initial resourceand knowledge stock, is determined by (15) and (37)-(38). The saddle path defines ηas a function of g, say ( ) ( ( ))η =t F g t . Substituting this and θ = θ into (37) and (38), we
16
find aggregate resource use, RT(t)+RY(t), as a function of g(t). Since we know theequation of motion for g, this solves for the entire extraction path. In equilibrium, g(0)must be such that cumulative extraction over the entire horizon exactly equals theinitial resource stock WR(0). As in the Cobb-Douglas case in section 4.1, the initialcondition n(0) has no effect on the growth rate. However, a higher initial resourcestock implies an equilibrium with a higher cost share η(0), and a lower growth rateg(0).
During the adjustment, the growth rate increases. This happens because, withrising resource prices and poor substitution in the T-sector, T-production becomesrelatively more expensive. Skilled labour moves from the T-sector to R&D. In thelong run, all skilled labour is in research so that that the asymptotic growth rate is
/S a irrespective of further model parameters.
Fig. 4(about here)
The main conclusion from this case is that poor substitution is in the knowledge-competing sector is not a problem for growth and investment in man-made(knowledge) capital. To the contrary: the rate of innovation in the steady state ishigher than in the case with unitary elasticities in both sectors, the case considered inthe previous section.
To draw conclusions about consumption growth rather than innovation, weneed to calculate again the rate of consumption growth. In the long run, the rate ofdepletion is again equal to the discount rate. However, depletion has a greaterweight in production in the case of poor substitution, since its share tends to one inthe long run, (1 ) 1− →η . Therefore, on the one hand the poor substitution case yieldshigher growth because of faster innovation, but on the other hand it yields a biggerdrag on growth through depletion. We can show that the former effect dominates thelatter, so that less substitution in the knowledge-competing sector implies higher long-rungrowth. Equation (43) still holds, provided η is replaced by 0η = . After substituting g= S/a and 0=η , we find the long-run growth rate of consumption for the case with σ= 1 > ω :
( ) ( )1, 1|ˆ 1 / 1 (1 ) (1 )SC
a= < = − + − − − + − σ ωθ β β θ δ φ φ θ φ ρ . (44)
This growth rate exceeds the growth rate of consumption with unitary elasticities inboth sectors, cf. (42)-(43), by the following positive amount:
1, 1 1, 1| |1ˆ ˆ (1 )(1 ) (1 (1 ) )
1 /SC Ca= < = =
− = + − − + − − +Φ σ ω σ ωρ φ η φ β θ
θ η.
4.4 Poor substitution in both consumer sectorsWe now turn to the general – and most interesting – case with elasticities unequalunity in both sectors. To be on the conservative side with respect to technologicalopportunities, we assume poor substitution, 0 1, 0 1< σ < < ω < , and small spillovers
17
to resource augmenting so that technological change is resource-using,(1 ) / 0= − − >ν β β δ . We have to examine the full system of three differential
equations (32), (31), and (34), the latter to be replaced by (35) in a corner solution.Solving for 0g η θ= = =!! ! with 0g ≥ , we can identify the different steady states.
From equation of motion (31), we see that θ always falls, so that in the steadystate θ must approach zero. Furthermore, from (32) we see that constancy of η in thesteady state requires either η = 0 or η = 1. Since η=1 can only be reached if ˆ 0>η attime infinity, and since ˆ 0>η violates the transversality condition (see (27) and (39)),any path converging to η = 1 must be ruled out. Hence, in the long run both θ and ηapproach zero:
( ) 0, ( ) 0∞ → ∞ →η θ . (45)
According to (34), the dynamics of g depend on the ratio θ/η. The growth rates of θand η approach asymptotically a (strictly negative) constant. Subtracting (32) from(31) and substituting (45), we find how the steady state ratio θ/η evolves over time:
1ˆ ˆ( ) ( ) ( )g− −∞ − ∞ = − ∞σ ω σθ η ρ νωσ σ
. (46)
Depending on parameters, three types of steady states arise: an interior solution, acorner solution with zero innovation, or a corner solution with maximal innovation.First consider for which value of θ(∞)/η(∞) an interior steady state, 0 < g(∞) < S/a, canarise. The inequality in (35) rules out θ(∞)/η(∞) → 0, since this would imply g = 0.Equation (34) rules out θ(∞)/η(∞)→∞ since this would imply 0g <! . Hence θ(∞)/η(∞)must be a constant in an interior steady state. This requires the both sides of equation(46) to be zero, so that ( ) ( ) / (1 )g ∞ = − −ρ σ ω ν σ ω. This solution is an interior solution,0 < g(∞) < S/a, only if ( ) ( ) / (1 )g ∞ = − −ρ σ ω ν σ ω< S/a, which can be reformulated as0 ( ) (1 ) /S a< − < −σ ω ρ σ ων . Second, consider the case σ < ω. We see from (46) thatthen ˆ ˆ( ) ( ) 0∞ − ∞ <θ η , so that θ(∞)/η(∞)→0, which implies, by (35), the corner solutiong = 0. Third, if 0 (1 ) / ( )S a< − < −σ ων σ ω ρ and g(∞) → S/a, we see from (46) thatˆ ˆ( ) ( ) 0∞ − ∞ >θ η so that θ(∞)/η(∞) → ∞. Equation (34) reveals that this is a steady state
( g! =0) provided ( / ) ( / )S a g⋅ −θ η approaches a bounded constant that is smaller than/ /S a+ρ ω . This is a rational expectations equilibrium as we show in the appendix,
where we use (34) to solve for the steady state value of ( / ) ( / )S a g⋅ −θ η in eachequilibrium.
Collecting these results we have:
( ) 0g ∞ = if σ ≤ ω. (47a)
( )( )(1 )
g ρ σ−ω∞ =ν −σ ω
if 0(1 )
Sa
σ−ω ν< <−σ ω ρ
(47b)
( )∞ → Sga
if (1 )
Sa
σ−ω ν≥−σ ω ρ
(47c)
18
The equations in (48) show that for given parameters, there is a unique steadystate. In the appendix we show the existence and stability of these steady states aswell as the transition paths to the steady states, which - together with the initialstocks n(0) and WR(0) - define the initial conditions for all endogenous variables. Theremainder of the section discusses the results and the implications for consumptiongrowth.
Equation (47b) reveals that innovation incentives keep intact and a constantinnovation rate can be maintained in the long run even with poor substitution,provided substitution in the knowledge-using Y-sector is better than in the R&D-competing T-sector (0 < ω < σ < 1). To understand this result, we have to sort out whythere is no incentive for skilled labour to move out of or into R&D in the long run.Two opposing – but inseparable – forces, from depletion and technological changerespectively, determine labour allocation. On the one hand, as the resource stock isdepleted and the amount of resource input per unit of labour falls, the wage falls,especially for the type of labour that is the poorest substitute for the resource. Thus, ifω < σ, the T-sector is hurt most by depletion and the relative wage of skilled labour,which is employed in this sector, falls. This lowers innovation costs and tends to raiseinnovation. On the other hand, any shift into R&D speeds up the pace of innovation,which makes capital goods relatively more abundant, lowers their reward (provided(1 ) 0− >σ ν , see (4)), and lowers the profits from innovation. On balance, in theinterior steady state (47b), innovation takes place at a rate that makes profits frominnovation fall at the same rate as costs of innovation (which happens because ofdepletion). The steady state with maximal R&D (47c) arises if the supply of skilledlabour is small since then the supply of skilled labour constrains the rate ofinnovation such that the depletion effect dominates: the relative wage paid by the T-sector falls even if asymptotically all skilled labour has moved out of the T-sector intoR&D. The zero innovation steady state (47a) arises if substitution is poorest in the Y-sector since then both depletion and innovation reduce the returns to innovation.
The interaction between depletion and innovation implies that the innovationrate becomes bounded by substitution elasticities when the supply of skilled labourgrows large. With ω < σ < 1, and S sufficiently large that (47b) applies, a rise in skilledlabour supply does not affect the innovation rate. Hence, the so-called scale effect, forwhich endogenous growth model were criticised (notably by Jones 1995, 1999), is notpresent. The reason is that the growth rate is determined by the equality of depletionand innovation bias effect, so that g is solely governed by technical and preferenceparameters, notably the elasticities (note that the scale effect is present in the casewith σ = 1, see section 4.1 and 4.3).
Another remarkable feature of the interior long-run innovation rate in (47b) isthat it rises with the discount rate. In the Cobb-Douglas case (section 4.1), and inmost endogenous growth models, the opposite happens. Usually, discountingdisfavours investment in general and investment in R&D in particular. However, inthe present model there are two types of investment, resource conservation andinnovation. Higher discounting reduces investment in the resources by speeding updepletion, see (40)-(41). Thus the wage of skilled labour in the T-sector falls relativelyfaster, which reduces the cost of R&D and speeds up innovation.
19
In the long run, with 0η θ= = growth of consumption is (equation (43) stillholds, provided η and θ are replaced by 0η = and 0=θ ):
C gφδ ρ= −
From this equation it becomes clear that growing consumption requires 0δ > , that is,endogenous knowledge has to affect the productivity of resource use in Y-production positively, or, in other words, technological change is resource-augmenting. In this case, long-run consumption growth is (technically) feasible inprinciple. However, our analysis shows that in the market equilibrium, consumptiongrows in the long run only if in addition to δ > 0, substitution is higher in theknowledge-using sector than in the knowledge-competing sector (σ > ω), and thediscount rate is low enough.
5. Conclusions
This paper shows that unbounded economic growth can be sustained if non-renewable resources are an essential input in production, even without exogenoustechnological change and with elasticities of substitution between man-made capitaland resources which lie below unity. We have used a multi-sector framework inwhich differences in substitution opportunities across sectors cause labour to movefrom production to R&D when the resource stock becomes depleted. Poorsubstitutability in the sector that competes for skilled labour input with the R&Dsector turns out to be favourable for growth. Resource depletion makes final goodsproduction activities that heavily rely on resources more expensive. Thus, increasedresource scarcity lowers the opportunity costs of innovation and shifts labour fromfinal goods production to innovation effort. The sectoral shift supplements inputchanges as a substitution mechanism. As a consequence, growth is higher with thiskind of poor substitutability compared to the case of unitary elasticities. In contrast,strong dependence on resources in the sector that implements the innovations is badfor growth: with a poor substitutability in this case, resource depletion increasesproduction costs and lowers the demand for innovations. We conclude that therelative resource dependence of the knowledge-using and knowledge-competingsectors (measured by cost shares and elasticities of substitution) determine whetherincentives for investment and innovation are sustained and growth is unbounded inthe presence of poor substitution possibilities. We also find that in the case of poorsubstitution, the size of the elasticities of substitution, rather than resource andlabour endowments, bound the rate of growth. Hence in the interior solution, thescale of the economy has no effect on long-run growth.
We have made some simplifying assumptions that may be relaxed in futureresearch. First, we have stressed that (in contrast to knowledge capital) physicalcapital inputs are bounded because material use is bounded. Instead of completelyabstracting from increases in the physical capital stock, physical capital accumulationcan be modelled subject to explicit material balances constraints. Second, we havemodelled technological change embodied in capital goods and we have found that ifresearch spillovers are large, technological change may become resource-saving.
20
Alternatively, we may model two types of innovation, one directed at improvingcapital productivity and the other at resource productivity. Third, we haveabstracted from resource extraction costs and polluting resource use, which may betaxed by the government. These features may change the price profile of the resourcebut they hit both consumer sectors in the same way. As the effects of price changes inthe two sectors work in opposite direction, as seen in sections 4.2 and 4.3, the qualityof our results is not expected to change substantially when enlarging the generalmodel set-up in this way. Fourth, to keep the set-up tractable we have assumed thatthere are two specific labour factors and that no innovation is possible in one of thesectors. The use of a single type of labour does not qualitatively change the results.The difference of the two sectors concerning knowledge use is an extreme form ofinput intensity which is not decisive for the outcome either. Finally, as the paperfocuses on market solutions, the issue of optimal policies has not been treated.Resource use produces no negative externalities in this model, only R&D generatespositive spillovers which leads, as in the original “Romer-type” approach to R&D, topositive subsidies for innovations in the social optimum.
References
Aghion, P. and P. Howitt (1998). Endogenous Growth Theory. Cambridge MA: MITPress.
Bovenberg A.L. and S. Smulders (1995). Environmental Quality and Pollution-augmenting Technological Change in a Two-sector Endogenous Growth Model,Journal of Public Economics 57, 369-391.
Bretschger, L. (1998). How to Substitute in order to Sustain: Knowledge DrivenGrowth under Environmental Restrictions. Environment and DevelopmentEconomics, 3, 861-893.
Cleveland, C. and M. Ruth (1997). When, where and by how much do biophysicallimits constrain the economic process; a survey of Nicolas Georgescu-Roegen'scontribution to ecological economics. Ecological Economics 22, 203-223.
Dasgupta, P.S. and G.M. Heal (1974). “The Optimal Depletion of ExhaustibleResources”, Review of Economic Studies, Symposium, 3-28.
Dasgupta, P.S., and G.M. Heal (1979) Economic Theory and Exhaustible Resources,Cambridge University Press.
Grimaud A. and L. Rougé (2003). “Non-renewable Resources and Growth withVertical Innovations: Optimum, Equilibrium and Economic Policies”, Journal ofEnvironmental Economics and Management 45, 433-453.
Grossman, G.M. and E. Helpman (1991). Innovation and Growth in the GlobalEconomy. Cambridge MA : MIT Press.
21
Groth, C. and P. Schou (2002): Can Non-renewable Resources Alleviate the Knife-edge Character of Endogenous Growth? Oxford Economic Papers 54, 386-411
Hartwick, J. M. (1977). Intergenerational Equity and the Investing of Rents fromExhaustible Resources, American Economic Review 67, 972-974.
Jones, C.I. (1995), “R&D based models of economic growth” Journal of PoliticalEconomy 103, 759-84.
Jones, C.I. (1999) “Growth: With or Without Scale Effects?” American EconomicReview 89, papers and proceedings, 139-144.
Meadows, D. H. et al. 1972. The Limits to Growth, New York: Universe Books.
Romer, P.M. (1990). Endogenous Technological Change, Journal of PoliticalEconomy, 98, s71-s102.
Scholz, C.M. and G. Ziemes (1999). Exhaustible Resources, MonopolisticCompetition, and Endogenous Growth, Environmental and Resource Economics 13,169-185.
Schou, P. (2000). Polluting Nonrenewable Resources and Growth. Environmental andResource Economics, 16, 211-227.
Solow, R.M. (1974) “Intergenerational Equity and Exhaustible Resources” Review ofEconomic Studies, Symposium, 29-45.
Stiglitz J.E. (1974). Growth with Exhaustible Natural Resources: Efficient andOptimal Growth Paths, Review of Economic Studies, 123-137.
22
Figures
Unskilled labourL
Differentiated capital components K
High-tech goodsY
Public knowledge
Research anddevelopment
Natural resourcesR
Standard goodsT
Skilled labourS
Homogenous capitalK
Y
T
Fig. 1
S Sa a
θφ ρη
−
*g Sa
g
Fig. 2
23
Fig. 3
Sa
0 1
0
0g
Fig. 4
0g =!↑
↓
↓←
←↑
θ
g
/S a
10
0=!θ←→
24
Appendix to section 4.4
This appendix studies existence and stability of the steady state with poorsubstitution in both sectors ( 0 1, 0 1< σ < < ω < , and 0ν > ). Because g! depends on
/θ η , see (34), whereas η = 0 in the steady state, see (45), we cannot directlydifferentiate the system in the steady state. Instead of studying the dynamics in termsof θ, η, and g, we therefore rewrite the system in terms of the three endogenousvariables g, η, and h, where h is defined as
Sh ga
θ = Φ − η . (A.1)
Existence of steady state with positive growthWe first examine a steady state with positive growth. Substituting / ( / )h S a gθ = η Φ −into from (34), (32) and (31), and subsequently setting 0θ = η = , see (45), we find thatthe following must hold in such a steady state:
0ρ = − ω + − = ω ! Sg g g h
a(A.2)
1ˆ 1h h g ρ −σ = − − + ν σ σ (A.3)
Moreover, if g > 0, 0η→ and 0θ → , the transversality condition boils down to (see(28), (33) and (39)):
ˆ 0Sr w h g− = − > (A.4)
From these three equations, it follows immediately that a positive growth raterequires /h g≤ +ρ ω, which rules out that h goes to infinity, which requires by (A.3)that [1 (1 ) / ] /h g≤ −µ − σ σ +ρ σ . The transversality condition rules out that h goes tozero, which requires by (A.3) that [1 (1 ) / ] /h g≥ −µ − σ σ +ρ σ . Hence, a positivegrowth rate requires constant h and from setting (A.3) equal zero we get
( ) ( ) [1 (1 ) / ] /h g∞ = ∞ ⋅ + ν −σ σ +ρ σ .
Substituting this solution into (A.2), and setting 0g =! , we find two solutions forthe innovation rate, corresponding to (47b) and (47c).
Existence of steady state without innovation.See main text.
Local stabilityWe prove that the steady state in the case of poor substitution in both sectors( 1, 1ω< σ < ) has two negative eigenvalues and one positive eigenvalue. We save on
25
notation by defining /s S a= . Provided that g is not at its corner g = 0, we may writefrom (32), (31), and (34):
(1 ) 1( )
( )1 (1 ) 1
( )
hs g
h h h g gh
s g
η− σ − Φ − = − − ρ − ρ + ν η − − σ − Φ −
!
(1 )(1 )( )h gη = −η − η −ω −! (A.5){ }( ) [ (1 ) ]( )g s g h g= − ρ − ω+ −ω η −!
The Jacobian of this system is:
/ / /( , , ) / / /
/ / /
h h h h gJ h g h g
g h g g g
∂ ∂ ∂ ∂η ∂ ∂ η ≡ ∂η ∂ ∂η ∂η ∂η ∂ = ∂ ∂ ∂ ∂η ∂ ∂
! ! !
! ! !! ! !
11 12 13
(1 )(1 ) (1 2 )(1 )( ) (1 )(1 )( )[ (1 ) ] ( )(1 )( ) ( )[ (1 ) ]
J J Jh g
s g s g h g h g s g
−η − η −ω − − η −ω − η − η −ω − − ω+ −ω η − − −ω − − + − ω+ −ω η −ρ
where
11 2
1( )(1 )(1 ) [ (1 ) ]
J h h g g − θ θ= + − −ρ − ρ + ν − σ − σ + − σ θ σ + − σ θ 2
12 2
1( )(1 )( ) [ (1 ) ]hJ gs g
= ρ + ν − σ Φ − σ + − σ θ
13 2
(1 )(1 ) 11 ( )(1 ) [ (1 ) ]
J h gs g
− σ − θ θ= − + ν − ρ + ν σ + − σ θ − σ + − σ θ
Interior growth rate, IGFirst, consider the steady state with 0 g s< < . In this case, we have
( / )[ (1 )] / (1 ) IGh h= ρ ω σ −ω+ ν − σ ν − σ ≡ , 0η = , ( / )( ) / (1 ) IGg g= ρ ω σ −ω ν − σ ≡ . Tofacilitate calculations, note also that [1 (1 ) / ] / /h g g= + ν − σ σ +ρ σ = +ρ ω. For thisequilibrium, all elements of the Jacobian J turn out to be finite, and the Jacobian canbe evaluated as:
12 [1 (1 ) / ]( ,0, ) 0 (1 ) / 0
( ) ( ) (1 ) / ( )
IG IG
IG IG
IG IG IG
h J hJ h g
s g s g s g
− + ν − σ σ = −ρ −ω ω − − ω − − ρ −ω ω − ω
We find that the determinant is positive:1Det ( ,0, ) (1 ) ( ) 0IG IG IG IGJ h g h s g− σ= ν −ωρ − >σ
26
Because the second row has zero elements only, except for the diagonal element, thediagonal element is an eigenvalue. Hence it follows immediately that one eigenvalueis negative:
,11
IG−ω λ = − ρ ω
.
Since the determinant of the Jacobian, which equals the product of the threeeigenvalues, is positive, we must have 1 2 3Det 0J = λ λ λ > , so there are two negativeand one positive eigenvalue.
2,2
2,3
1 1[ ( )] [ ( )] 4 ( ) (1 ) / 02 21 1[ ( )] [ ( )] 4 ( ) (1 ) / 02 2
IG IG IG IG IG IG IG
IG IG IG IG IG IG IG
h s g h s g s g h
h s g h s g s g h
λ = +ω − − +ω − + ω − ν − σ σ <
λ = +ω − + +ω − + ω − ν − σ σ >
Maximum growth rate, MGNext consider the steady state equilibrium with g s→ . For this case, we evaluate theJacobian at [1 (1 ) / ] / MGh s h= + ν − σ σ + ρ σ ≡ , 0η = , g s= . To facilitate calculations,note that it implies 0θ = , [1 (1 ) / ] / /h g g= + ν − σ σ +ρ σ < +ρ ω. For this equilibrium,J12 and J13 cannot be evaluated because they involve a division by zero. However, thedeterminant and the characteristic equation can be determined since multiplyingelements from the first row with elements from the third row always gives finiteexpressions. In particular, the determinant and characteristic equation can be writtenas:
Det ( ,0, ) (1 )( )[ ( )] 0MG MG MG MGJ h s h h s h s= −ω − ρ−ω − >0 [(1 )( ) ] [ ( ) ] ( )MG MG MGh s h s h= −ω − + λ ⋅ ρ −ω − + λ ⋅ − λ
Hence, the eigenvalues are
,1
,2
,3
(1 )( ) 0,[ ( )] 0
0
MG MG
MG MG
MG MG
h sh s
h
λ = − −ω − <λ = − ρ−ω − <
λ = >
Zero growth rate, ZGFinally, consider the steady state with 0g g= =! . The dynamics are now governed by(32) and (31) (note that we cannot use the system in (A.5), which is valid for g > 0only). The Jacobian, evaluated at the steady state with 0gη = θ = = reads:
/ / (1 ) / 0/ / 0 (1 ) /
∂η ∂η ∂η ∂θ −ρ −ω ω = ∂θ ∂η ∂θ ∂θ −ρ − σ σ
! !! !
So the two eigenvalues are negative:
27
,1
,2
1
1
ZG
ZG
−ω λ = − ρ ω − σ λ = − ρ σ
Adjustment and initial conditionsTwo negative eigenvalues are associated with each steady state. This implies that theinitial values of (g, η, h) have to be located on the two-dimensional stable manifold,spanned by the eigenvectors associated with the two negative eigenvalues. In thelinearised version, we can identify the initial condition as a the intersection betweenthe manifold and two planes to be defined as follows. First, the marginal product ofresource use has to be equalized across the two sectors, as described in equations(36)-(38). By eliminating RY and RT between these equations, we find a relationbetween g, η, θ and n, say J(g, η, θ, n)=0, which – after substitution of the definition ofh (A.1) – defines a plane in the (g, η, h) space. Second, the resource stock has to beasymptotically depleted, as described in equation (15). Substituting (36)-(37) into (15)and integrating, we find a relation between g, η, θ, n, and WR, say J(g, η, θ, n, WR)=0,which – after substitution of the definition of h – defines a plane in the (g, η, h) space.The intersection between the two planes, defined for initial values n(0) and WR(0),and the stable manifold determines the initial equilibrium.
Bretschger and SmuldersSustainability and substitution of exhaustible natural resources
Some notes on the derivations of the equations in the main text.
Derivation of (4) and (6), relative factor demand.The producer of Y-goods maximizes
1 11
0 0
(1 )( )n n
Y Y j Y R Y kj jp k dj n R p R p k dj
σσ− σ−
σ−βσβ δ σ
π = θ + − θ − − ∫ ∫ (B.1)
The first order conditions are:
1 111/
0
0n
YY j j kj
j
p Y k dj k pk
σ− −βσ
β β−σ ∂π= ⇔ θ = ∂
∫ (B.2)
1/1/ ( 1) /0 (1 )YY Y R
Y
p Y n R pR
− σσ δ σ− σ∂π= ⇔ − θ =
∂(B.3)
Inserting the symmetry result, /j Yk K n= , we can rewrite (B.2) as:
1/1/ (1 )( 1) /Y Y kYp Y n K p− σσ −β σ− βσθ = . (B.4)
Dividing (B.3) by (B.4), we find (4).Equation (6) is derived analogously.
Derivation of (7), monopoly price of capital.The monopolistic supplier of kj maximizes ( ) Lp k k w kπ = − , where we omit the subscript jand where p(k) is the inverse demand function for capital, defined by the second equality in(4.2). The first order condition with respect to k reads: (1 '( ) / ) 0Lp p k k p w+ − = . From(4.2) we see that '( ) / 1p k k p = β− if we make the standard assumption that the producerignores its own influence on aggregate demand (Dixit and Stiglitz ,1977). Substituting thisresult, we find (7).
Derivation of (20),(21),(22), consumer demandThe representative household maximizes (18)-(19), subject to the budget constraint
L S T YN rN w L w S p T p Y= + + − −! , where N = npn is asset holdings. The Hamiltonianreads:
(13.1) log (1 ) log [ ]L S T YH Y T rN w L w S p T p Y= φ + −φ + λ + + − − (B.5)
where λ is the co-state variable. The conditions for a maximum are:
0 / 0YH Y pY∂ = ⇔ φ − λ =∂
, (B.6)
0 (1 ) / 0TH T pY∂ = ⇔ −φ − λ =∂
, (B.7)
H rN∂ = ρλ − λ ⇔ λ = ρλ − λ∂
! ! . (B.8)
lim 0t
tNe−ρ
→∞λ = . (B.9)
Eliminating λ between (B.6) and (B.7), we find (20). Solving for λ in (B.7), we findˆ ˆ(1 ) /( ) ( )T TTp T pλ = − φ ⇔ λ = λ ⋅ +! . Substituting these results in (B.8), we find (21).
Substituting (B.8) and nN np= into (B.9), we find (22).
Derivation of (26) and (27), factor shares in terms of prices.By the definitions in (23) and (24), the capital and resource share in the Y-sector are
( ) /( )kY Y Yp K p Yθ = ⋅ ⋅ and 1 ( ) /( )R Y Yp R p Y− θ = ⋅ ⋅ , respectively, the capital andresource share in the T-sector are ( ) /( )KT T Tp K p Tη = and 1 ( ) /( )R T Tp R p T− η = ,respectively. Hence, (4) and (6) can be written as:
1
1 1kY
R
pn p
σ−σ
ν
θ θ= − θ − θ , (B.10)
1
1 1KT
R
pp
ω−ω η η= ⋅ − η − η , (B.11)
or, expressed in percentage changes:
1ˆ ˆ ˆ ˆ(1 ) (1 )( )kY Rp n p−θ − θ = − σ − ν − , (B.12)
( )( )1ˆ ˆ ˆ(1 ) 1 KT Rp p−η − η = −ω − . (B.13)
where hats denote growth rates. Substituting (7), (10), (13) and (25) to eliminateˆ ˆ ˆ ˆ, , ,L S Rw w p n respectively, we find (26) and (27).
Derivation of (31) and (32), factor shares in terms of innovation rate.Differentiating (28) with respect to time, we find:
ˆ ˆˆˆ ˆL S Tw w K− = θ − η+ . (B.14)
Substituting (29) to eliminate ˆTK , we find:
ˆˆ Lr w− =ρ− θ. (B.15)
Substituting this result into (26), we find (31).Differentiating (30) with respect to time, we find:
ˆ/T
gKS a g
= −−!
. (B.16)
Substituting (29) into (27) to eliminate ˆ Sr w− , we find:
ˆˆ ˆ(1 )(1 )( )TKη = − −ω − η ρ+ − η . (B.17)
Substituting (B.16) into (B.17), and rewriting, we find (32).
Deriving (36) and (37), rates of depletion.Multiplying both sides of (4) by ( / )Y YK R −σ , we find:
1(1 )
1kYY Y
Y R Y
pK K nR p R
σ−σ −σ− −σ ν θ= − θ
(B.18)
Inserting the definition of θ from (23), we find:
1(1 )
11Y
Y
K nR
σ−σ −σ− −σ ν θ θ = − θ− θ
(B.18)
Using (17) to eliminate KY, and rewriting, we find (36).Multiplying both sides of (6) by ( / )T TK R −ω and inserting the definition of η from (24), wefind:
1
11T
T
KR
ω−ω −ω η η= − η− η (B.19)
Using (30) to eliminate KT, and rewriting, we find (37).
